Multiple Objects, Multiple UV's and Multiple Shape Keys Conundrum
Select all your objects (make sure all objects have the correct shape key selected and the correct uv set go active) and press Tab to enter Edit Mode for all objects at once. Then you can edit all the UVs all at once.

You should also think about the reason you got into a big mess where you would need to do that and if there is another way. If not, here is a simple script template to select all the shapekeys and all the uvs at once. if the UV is not present yet it will be created.

import bpy
UV_name = "Your UV Name"
for o in bpy.context.selected_objects:
    o.active_shape_key_index = 0
    if UV_name not in o.data.uv_layers:
        o.data.uv_layers.new(name=UV_name)
    o.data.uv_layers[UV_name].active = true